Print ViewElectric Potential Energy versus Electric Potential

1. The problem statement, all variables and given/known dataLearning Goal: To understand the relationship and differences between electric potential and electric potential energy.

In this problem we will learn about the relationships between electric force [tex]\vec{F}[/tex], electric field [tex]\vec{E}[/tex], potential energy U, and electric potential V. To understand these concepts, we will first study a system with which you are already familiar: the uniform gravitational field.

Part 9
The electric field can be derived from the electric potential, just as the electrostatic force can be determined from the electric potential energy. The relationship between electric field and electric potential is [tex]\vec{E} = -\vec{\nabla}V[/tex], where [tex]\vec{\nabla}[/tex] is the gradient operator:

Consider again the electric potential [tex]V=-Ez[/tex] corresponding to the field [tex]\vec{E}=E\hat{z}[/tex]. This potential depends on the z coordinate only, so [tex]\frac{\partial V}{\partial x}=\frac{\partial V}{\partial y}=0[/tex] and [tex]\frac{\partial V}{\partial z}=\frac{dV}{dz}[/tex].

Find an expression for the electric field [tex]\vec{E}[/tex] in terms of the derivative of [tex]V[/tex].
Express your answer as a vector in terms of the unit vectors [tex]\hat{x}[/tex], [tex]\hat{y}[/tex], and/or [tex]\hat{z}[/tex]. Use [tex]dV/dz[/tex] for the derivative of [tex]V[/tex] with respect to [tex]z[/tex].